Cette thèse est consacrée à l'étude de l'influence d'une fine couche hétérogène sur le comportement élastique linéaire d'une structure tridimensionnelle.Deux types d'hétérogénéités sont pris en compte : des cavités et des inclusions élastiques. Une étude complémentaire, dans le cas d'inclusions de grande rigidité, a été réalisée en considérant un problème de conduction thermique.Une analyse formelle par la méthode des développements asymptotiques raccordés conduit à un problème d'interface qui caractérise le comportement macroscopique de la structure. Le comportement microscopique de la couche est lui déterminé sur une cellule de base. Le modèle asymptotique obtenu est ensuite implémenté dans un code éléments finis. Une étude numérique permet de valider les résultats de l'analyse asymptotique. / This thesis is devoted to the study of the influence of a thin heterogeneous layeron the linear elastic behavior of a three-dimensional structure. Two types of heterogeneties are considered : cavities and elastic inclusions. For inclusions of high rigidty a further study was performed in the case of a heat conduction problem.A formal analysis using the matched asymptotic expansions method leads to an interface problem which characterizes the macroscopic behavior of the structure. The microscopic behavior of the layer is determined in a basic cell.The asymptotic model obtained is then implemented in a finite element software.A numerical study is used to validate the results of the asymptotic analysis.

Abstract Several numerical methods and algorithms, for solving the mathematical model of a continuous casting process, are presented, and theoretically studied, in this work. The numerical algorithms can be divided in to three different groups: the Schwarz type overlapping methods, the nonoverlapping Splitting iterative methods, and the Predictor-Corrector type nonoverlapping methods. These algorithms are all so-called parallel algorithms i.e., they are highly suitable for parallel computers. Multiplicative, additive Schwarz alternating method and two asynchronous domain decomposition methods, which appear to be a two-stage Schwarz alternating algorithms, are theoretically and numerically studied. Unique solvability of the fully implicit and semi-implicit finite difference schemes as well as monotone dependence of the solution on the right-hand side are proved. Geometric rate of convergence for the iterative methods is investigated. Splitting iterative methods for the sum of maximal monotone and single-valued monotone operators in a finite-dimensional space are studied. Convergence, rate of convergence and optimal iterative parameters are derived. A two-stage iterative method with inner iterations is analyzed in the case when both operators are linear, self-adjoint and positive definite. Several new finite-difference schemes for a nonlinear convection-diffusion problem are constructed and numerically studied. These schemes are constructed on the basis of non-overlapping domain decomposition and predictor-corrector approach. Different non-overlapping decompositions of a domain, with cross-points and angles, schemes with grid refinement in time in some subdomains, are used. All proposed algorithms are extensively numerically tested and are founded stable and accurate under natural assumptions for time and space grid steps. The advantages and disadvantages of the numerical methods are clearly seen in the numerical examples. All of the algorithms presented are quite easy and straight forward, from an implementation point of view. The speedups show that splitting iterative method can be parallelized better than multiplicative or additive Schwarz alternating method. The numerical examples show that the multidecomposition method is a very effective numerical method for solving the continuous casting problem. The idea of dividing the subdomains to smaller subdomains seems to be very beneficial and profitable. The advantages of multidecomposition methods over other methods is obvious. Multidecomposition methods are extremely quick, while being just as accurate as other methods. The numerical results for one processor seem to be very promising.

In order to prepare modern finite element analysis a program for the efficient parallel generation of tetrahedral meshes in a wide class of three dimensional domains having a generalized cylindric shape is presented. The applied mesh generation strategy is based on the decomposition of some 2D-reference domain into single con- nected subdomains by means of its triangulations the tetrahedral layers are built up in parallel. Adaptive grid controlling as well as nodal renumbering algorithms are involved. In the paper several examples are incorporated to demonstrate both program's capabilities and the handling with.

A parallel realisation of the Conjugate Gradient Method with Schur-Complement preconditioning, based on a domain decomposition approach, is described in detail. Special kinds of solvers for the resulting interiour and coupling systems are presented. A large range of numerical results is used to demonstrate the properties and behaviour of this solvers in practical situations.

This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

Cette thèse s’inscrit dans le cadre de l’étude du post-flambement local des grandes struc- tures composites raidies. La simulation du post- flambement par la méthode des éléments- finis est aujourd’hui limitée par le coût du calcul en particulier pour les grandes structures. Seules des zones restreintes peuvent être étudiées, en négligeant les interactions global/local. L’objectif de cette thèse est de proposer une stratégie de calcul performante pour la simula- tion du post-flambement local des grandes structures raidies à partir des connaissances sur le comportement mécanique des structures en post-flambement et d’un découpage naturel le long des raidisseurs favorable au calcul parallèle.Dans la littérature, les méthodes de réduction de modèle adaptative ont démontré leur ca- pacité à réduire le nombre d’inconnues tout en maîtrisant l’erreur d’approximation de la solution des problèmes non-linéaires. Par ailleurs, les méthodes de décomposition de do- maine avec localisation non-linéaire sont particulièrement adaptées au calcul parallèle en mécanique des structures en présence de non-linéarités locales.Les travaux de thèse portent dans un premier temps sur une stratégie de réduction de modèle adaptative spécifique au cas du post-flambement. Dans le cas d’un flambement local d’une grande structure raidie une combinaison avec une méthode de décomposition de domaine primale est ensuite proposée. Toutes ces stratégies sont implémentées dans un code de re- cherche programmé pendant la thèse. / This thesis is part of the study of local post-buckling of large stiffened composite struc- tures. The finite element simulation of structures subjected to post-bucking still faces com- putational limits, especially for large structures. Only restricted area may be studied for now, neglecting global/local interactions.The aim of the thesis is to propose an efficient computational strategy for local post-buckling analysis of large stiffened structures from knowledge on mechanical behavior of post-buckling structures and a natural partitionning along stiffeners conducive to parallel computation. In litterature, the adaptive model reduction solving techniques have demonstrated their abi- lity to drastically reduce the number of unknowns as well as to control the approximation error of solving non-linear problems. Furthermore, domain decomposition methods with a non-linear local step are suited to parallel computation in structural mechanics in the pre- sence of local non-linearities.Our work deals first with an adaptive model reduction strategy dedicated to post-buckling problems. In order to adress larger stiffened structures subjected to local post-buckling, like an aircraft fuselage, partitioning is then performed. The model reduction, as well as the adap- tive procedure are written in the framework of the primal domain decomposition method with a non-linear local step. These strategies are implemented in a research code developed for the purpose of the thesis.

L'objectif de cette thèse est de concevoir des méthodes de décomposition de domaine qui sont robustes même pour les problèmes difficiles auxquels on est confronté lorsqu'on simule des objets industriels ou qui existent dans la nature. Par exemple une difficulté à laquelle est confronté Michelin et que les pneus sont constitués de matériaux avec des lois de comportement très différentes (caoutchouc et acier). Ceci induit un ralentissement de la convergence des méthodes de décomposition de domaine classiques dès que la partition en sous domaines ne tient pas compte des hétérogénéités. Pour trois méthodes de décomposition de domaine (Schwarz Additif, BDD et FETI) nous avons prouvé qu¿en résolvant des problèmes aux valeurs propres généralisés dans chacun des sous domaines on peut identifier automatiquement quels sont les modes responsables de la convergence lente. En d¿autres termes on divise le problème de départ en deux : une partie où on peut montrer que la méthode de décomposition de domaine va converger et une seconde où on ne peut pas. L¿idée finale est d¿appliquer des projections pour résoudre ces deux problèmes indépendemment (c¿est la déflation) : au premier on applique la méthode de décomposition de domaine et sur le second (qu¿on appelle le problème grossier) on utilise un solveur direct qu¿on sait être robuste. Nous garantissons théorétiquement que le solveur à deux niveaux qui résulte de ces choix est robuste. Un autre atout de nos algorithmes est qu¿il peuvent être implémentés en boite noire ce qui veut dire que les matériaux hétérogènes ne sont qu¿un exemple des difficultés qu¿ils peuvent contourner / The objective of this thesis is to design domain decomposition methods which are robust even for hard problems that arise when simulating industrial or real life objects. For instance one particular challenge which the company Michelin is faced with is the fact that tires are made of rubber and steel which are two materials with very different behavior laws. With classical domain decomposition methods, as soon as the partition into subdomains does not accommodate the discontinuities between the different materials convergence deteriorates. For three popular domain decomposition methods (Ad- ditive Schwarz, FETI and BDD) we have proved that by solving a generalized eigenvalue problem in each of the subdomains we can identify automatically which are the modes responsible for slow convergence. In other words we can divide the original problem into two problems : the first one where we can guarantee that the domain decomposition method will converge quickly and the second where we cannot. The final idea is to apply projections to solve these two problems independently (this is also known as deflation) : on the first we apply the domain decomposition method and on the second (we call it the coarse space) we use a direct solver which we know will be robust. We guarantee theoretically that the resulting two level solver is robust. The other main feature of our algorithms is that they can be implemented as black box solvers meaning that heterogeneous materials is only one type of difficulty that they can identify and circumvent.

Introduction In recent years, domain decomposition methods have been used extensively to efficiently solve boundary value problems for partial differential equations in complex{form domains. On the other hand, multilevel techniques on hierarchical data structures also have developed into an effective tool for the construction and analysis of fast solvers. But direct realization of multilevel techniques on a parallel computer system for the global problem in the original domain involves difficult communication problems. I this paper, we present and analyze a combination of these two approaches: domain decomposition and multilevel decomposition on hierarchical structures to design optimal preconditioning operators.

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Ces travaux de recherche ont pour objectif de contribuer au développement et à l'exploitation d'outils de vérification des problèmes de mécanique linéaires dans le cadre des méthodes de décomposition de domaine sans recouvrement. Les apports de cette thèse sont multiples : * Nous proposons d'améliorer la qualité des champs statiquement admissibles nécessaires à l'évaluation de l'estimateur par une nouvelle méthodologie de reconstruction des contraintes en séquentiel et par des optimisations du calcul de l'intereffort en cadre sous-structuré.* Nous démontrons des bornes inférieures et supérieures de l'erreur séparant l'erreur algébrique (due au solveur itératif) de l'erreur de discrétisation (due à la méthode des éléments finis) tant pour une mesure globale que pour une quantité d'intérêt. Cette séparation permet la définition d'un critère d'arrêt objectif pour le solveur itératif.* Nous exploitons les informations fournies par l'estimateur et les espaces de Krylov générés pour mettre en place une stratégie auto-adaptative de calcul consistant en une chaîne de résolution mettant à profit remaillage adaptatif et recyclage des directions de recherche. Nous mettons en application le pilotage du solveur par un objectif de précision sur des exemples mécaniques en deux dimensions. / This research work aims at contributing to the development of verification tools in linear mechanical problems within the framework of non-overlapping domain decomposition methods.* We propose to improve the quality of the statically admissible stress field required for the computation of the error estimator thanks to a new methodology of stress reconstruction in sequential context and thanks to optimizations of the computations of nodal reactions in substructured context.* We prove guaranteed upper and lower bounds of the error that separates the algebraic error (due to the iterative solver) from the discretization error (due to the finite element method) for both global error measure mentand goal-oriented error estimation. It enables the definition of a new stopping criterion for the iterative solver which avoids over-resolution.* We benefit the information provided by the error estimator and the Krylov subspaces built during the resolution to set an auto-adaptive strategy. This strategy consists in sequel of resolutions and takes advantage of adaptive remeshing and recycling of search directions .We apply the steering of the iterative solver by objective of precision on two-dimensional mechanical examples.